Cyclicity of bicyclic operators and completeness of translates

We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if S: (a_n) _nεℤ→ a_n-1nεℤ is the shift operator acting on the weighted space of sequences ℓ²ℤ , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if limⁿ→ +rm log}ω(-n){n}=0} . On the other hand one can see that S is not cyclic if the series ∑_ngeq 1} logω (-n)}/n² diverges. We show that the question of Herrero whether either S or S* is cyclic on ℓω² ℤ admits a positive answer when the series ∑_nεℤ rm log} ^S{n}||/(n²+1)} is convergent. We also prove completeness results for translates in certain Banach spaces of functions on ℝ.